Coverability of Reset Petri Nets and Other Well-Structured Transition Systems by Partial Deduction

  • Michael Leuschel
  • Helko Lehmann
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1861)


In recent work it has been shown that infinite state model checking can be performed by a combination of partial deduction of logic programs and abstract interpretation. It has also been shown that partial deduction is powerful enough to mimic certain algorithms to decide coverability properties of Petri nets. These algorithms are forward algorithms and hard to scale up to deal with more complicated systems. Recently, it has been proposed to use a backward algorithm scheme instead. This scheme is applicable to so-called well-structured transition systems and was successfully used, e.g., to solve coverability problems for reset Petri nets. In this paper, we discuss how partial deduction can mimic many of these backward algorithms as well. We prove this link in particular for reset Petri nets and Petri nets with transfer and doubling arcs. We thus establish a surprising link between algorithms in Petri net theory and program specialisation, and also shed light on the power of using logic program specialisation for infinite state model checking.


Model Check Logic Program Logic Programming Decision Procedure Partial Evaluation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    P. A. Abdulla, K. Čerāns, B. Jonsson, and Y.-K. Tsay. General decidability theorems for infinite-state systems. In Proceedings LICS’96, pages 313–321, July 1996. IEEE Computer Society Press.Google Scholar
  2. 2.
    T. Araki and T. Kasami. Some decision problems related to the reachability problem for Petri nets. Theoretical Computer Science, 3:85–104, 1977.zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    B. Bérard and L. Fribourg. Reachability analysis of (timed) petri nets using real arithmetic. In Proceedings Concur’99, LNCS 1664, pages 178–193. Springer-Verlag, 1999.Google Scholar
  4. 4.
    W. Charatonik and A. Podelski. Set-based analysis of reactive infinite-state systems. In B. Steffen, editor, Proceedings TACAS’98, LNCS 1384, pages 358–375. Springer-Verlag, March 1998.Google Scholar
  5. 5.
    D. De Schreye, R. Glück, J. Jørgensen, M. Leuschel, B. Martens, and M. H. Sørensen. Conjunctive partial deduction: Foundations, control, algorithms and experiments. J. Logic Progam., 41(2 & 3):231–277, November 1999.Google Scholar
  6. 6.
    C. Dufourd, A. Finkel, and P. Schnoebelen. Reset nets between decidability and undecidability. In Proceedings ICALP’98, LNCS 1443, pages 103–115. Springer-Verlag, 1998.Google Scholar
  7. 7.
    J. Ezparza. Decidability of model-checking for infinite-state concurrent systems. Acta Informatica, 34:85–107, 1997.CrossRefMathSciNetGoogle Scholar
  8. 8.
    A. Finkel. The minimal coverability graph for Petri nets. Advances in Petri Nets 1993, LNCS 674, pages 210–243, 1993.Google Scholar
  9. 9.
    A. Finkel and P. Schnoebelen. Fundamental structures in well-structured infinite transition systems. In Proceedings LATIN’98, LNCS 1380, pages 102–118. Springer-Verlag, 1998.Google Scholar
  10. 10.
    A. Finkel and P. Schnoebelen. Well-structured transition systems everywhere ! Theoretical Computer Science, 2000. To appear.Google Scholar
  11. 11.
    L. Fribourg and H. Olsen. Proving Safety Properties of Infinite State Systems by Compilation into Presburger Arithmtic. In Proceedings Concur’97, LNCS 1243, pages 213–227. Springer-Verlag, 1997.Google Scholar
  12. 12.
    R. Glück and M. Leuschel. Abstraction-based partial deduction for solving inverse problems-a transformational approach to software verification. In Proceedings PSI’99, LNCS 1755, pages 93–100, 1999. Springer-Verlag.Google Scholar
  13. 13.
    B. Heinemann. Subclasses of self-modifying nets. In Applications and Theory of Petri Nets, pages 187–192. Springer-Verlag, 1982.Google Scholar
  14. 14.
    T. A. Henzinger and P.-H. Ho. HYTECH: The Cornell HYbrid TECHnology tool. Hybrid Systems II, LNCS 999:265–293, 1995.Google Scholar
  15. 15.
    R. M. Karp and R. E. Miller. Parallel program schemata. Journal of Computer and System Sciences, 3:147–195, 1969.zbMATHMathSciNetGoogle Scholar
  16. 16.
    L. Lafave and J. Gallagher. Constraint-based partial evaluation of rewriting-based functional logic programs. In N. Fuchs, editor, Proceedings LOPSTR’97, LNCS 1463, pages 168–188, July 1997.Google Scholar
  17. 17.
    J.-L. Lassez, M. Maher, and Marriott. Unification revisited. In J. Minker, editor, Foundations of Deductive Databases and Logic Programming, pages 587–625. Morgan-Kaufmann, 1988.Google Scholar
  18. 18.
    M. Leuschel. Program specialisation and abstract interpretation reconciled. In J. Jaffar, editor, Proceedings JICSLP’98, pages 220–234, Manchester, UK, June 1998. MIT Press.Google Scholar
  19. 19.
    M. Leuschel. Logic program specialisation. In J. Hatcliff, T.Æ. Mogensen, and P. Thiemann, editors, Partial Evaluation: Practice and Theory, LNCS 1706, pages 155–188 and 271–292, 1999. Springer-Verlag.Google Scholar
  20. 20.
    M. Leuschel and D. De Schreye. Logic program specialisation: How to be more specific. In H. Kuchen and S. Swierstra, editors, Proceedings PLILP’96, LNCS 1140, pages 137–151, September 1996. Springer-Verlag.Google Scholar
  21. 21.
    M. Leuschel and D. De Schreye. Constrained partial deduction and the preservation of characteristic trees. New Gen. Comput., 16:283–342, 1998.CrossRefGoogle Scholar
  22. 22.
    M. Leuschel and H. Lehmann. Solving Coverability Problems of Petri Nets by Partial Deduction. Submitted.Google Scholar
  23. 23.
    M. Leuschel, B. Martens, and D. De Schreye. Controlling generalisation and polyvariance in partial deduction of normal logic programs. ACM Transactions on Programming Languages and Systems, 20(1):208–258, January 1998.Google Scholar
  24. 24.
    M. Leuschel and T. Massart. Infinite state model checking by abstract interpretation and program specialisation. In A. Bossi, editor, Proceedings LOPSTR’99, LNCS 1817, pages 63–82, Venice, Italy, September 1999.Google Scholar
  25. 25.
    J. W. Lloyd and J. C. Shepherdson. Partial evaluation in logic programming. J. Logic Progam., 11(3& 4):217–242, 1991.CrossRefMathSciNetzbMATHGoogle Scholar
  26. 26.
    K. Marriott, L. Naish, and J.-L. Lassez. Most specific logic programs. Annals of Mathematics and Artificial Intelligence, 1:303–338, 1990.zbMATHCrossRefGoogle Scholar
  27. 27.
    B. Martens and J. Gallagher. Ensuring global termination of partial deduction while allowing flexible polyvariance. In L. Sterling, editor, Proceedings ICLP’95, pages 597–613, June 1995. MIT Press.Google Scholar
  28. 28.
    E. W. Mayr. An algorithm for the general Petri net reachability problem. Siam Journal on Computing, 13:441–460, 1984.zbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Y. S. Ramakrishna, C. R. Ramakrishnan, I. V. Ramakrishnan, S. A. Smolka, T. Swift, and D. S. Warren. Efficient model checking using tabled resolution. In Proceedings CAV’97, LNCS 1254, pages 143–154. Springer-Verlag, 1997.Google Scholar
  30. 30.
    W. Reisig. Petri Nets-An Introduction. Springer Verlag, 1982.Google Scholar
  31. 31.
    J. Rushby. Mechanized formal methods: Where next? In Proceedings of FM’99, LNCS 1708, pages 48–51, Sept. 1999. Springer-Verlag.Google Scholar
  32. 32.
    M. H. Sørensen. Convergence of program transformers in the metric space of trees. In Proceedings MPC’98, LNCS 1422, pages 315–337. Springer-Verlag, 1998.Google Scholar
  33. 33.
    M. H. Sørensen and R. Glück. An algorithm of generalization in positive supercompilation. In J. W. Lloyd, editor, Proceedings ILPS’95, pages 465–479, December 1995. MIT Press. Verlag, 1978.Google Scholar
  34. 35.
    R. Valk and G. Vidal-Naquet. Petri nets and regular languages. Journal of Computer and System Sciences, 23(3):299–325, Dec. 1981.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Michael Leuschel
    • 1
  • Helko Lehmann
    • 1
  1. 1.Department of Electronics and Computer ScienceUniversity of SouthamptonHighfieldUK

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